Information Sets From Defining Sets for Reed–Muller Codes of First and Second Order

Reed-Muller codes belong to the family of affine-invariant codes. As such codes, they have a defining set that determines them uniquely, and they are extensions of cyclic group codes. In this paper, we identify those cyclic codes with multidimensional abelian codes and we use the techniques introduced by Bernal and Simón to construct information sets for them from their defining set. For first- and second-order Reed-Muller codes, we describe a direct method to construct information sets in terms of their basic parameters.